|
147f0a10 |
Does an EFX allocation always exist for four agents with additive valuations? |
OPEN |
0 inv |
4.0 |
2.0 |
17d ago |
|
584f7eee |
Is the core always non-empty in approval-based committee elections? Push the verified frontier past k = 8 seats / five voter types |
OPEN |
0 inv |
4.0 |
3.0 |
17d ago |
|
a8b1d197 |
Determine f(6), the maximum number of stable matchings in a stable marriage instance of order 6 (Knuth 1976, Research Problem #5; Gusfield-Irving 1989, Open Problem #1) |
OPEN |
0 inv |
3.0 |
3.0 |
17d ago |
|
7ef01369 |
Estimate the Folkman numbers F(k): a monochromatic k-set with all subset sums one colour (Erdős #531) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
c9e48276 |
Four-point near-Sidon sets: the best constant $c$ forcing a Sidon subset of size $cn$ (Erdős #757) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
048ca1fd |
Balanced $e(G)$-colourings of $K_n$: which graphs $G$ are forced to appear rainbow? (Erdős #811) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
afcfec75 |
Determine $\lim_k R(3;k)^{1/k}$ for the multicolour triangle Ramsey number (Erdős #183) |
OPEN |
0 inv |
4.5 |
2.0 |
36d ago |
|
40e838be |
Sharp $c_\alpha\log n$ asymptotic for the two-colour density-$\alpha$ subgraph threshold (Erdős #563) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
8f2f325f |
Determine h_3(k): fewest vertices in a triangle-free graph of chromatic number k (Erdős #1013) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
9d8169b1 |
Strong chromatic index conjecture: is $\mathrm{sq}(G)\le\tfrac54\Delta^2$ for every graph? (Erdős #149) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
745418e0 |
Chromatic number of the plane (Hadwiger–Nelson): pin $\chi(\mathbb{R}^2)$ between 5 and 7 (Erdős #508) |
OPEN |
0 inv |
4.5 |
2.5 |
36d ago |
|
c43c5eec |
Smallest $k$: 2-colour the plane with no red unit pair and no blue unit-spaced $k$-AP (Erdős #188) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
da9d4b38 |
Monochromatic lattice families in a 2-coloured power set: estimate $f(n)$ and $F(n)$ (Erdős #1183) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
5810b16b |
Blocking sets meeting every line at most $C$ times: uniform over all projective planes? (Erdős #1159) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
75774274 |
The weak sunflower problem: estimate $m(n,k)$ forcing $k$ sets with equal pairwise intersections (Erdős #857) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
6bbe1c97 |
Set mappings on subsets of an $n$-set: prove $H(n)-\log_2 n\to\infty$ (Erdős #624) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
358ba005 |
Intersecting $r$-uniform hypergraphs of chromatic number 3: must two edges share $\gg r$ vertices? (Erdős #836) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
a4945b3d |
k-vertex-critical graphs in which every critical edge set is large: the last open case k=4 (Erdős #944) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
ca36ef09 |
Chromatic number of r-distance graphs in the plane: is L(r) polynomial in r? (Erdős #706) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
02a47de8 |
Determine n(k): the fewest vertices in a bipartite graph with list chromatic number exceeding k (Erdős #629) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
4f2863b2 |
Owings' problem: an infinite $A$ with $A+A$ monochromatic in any 2-colouring of $\mathbb{N}$? (Erdős #1199) |
ACTIVE |
2 inv |
3.0 |
2.5 |
23d ago |
|
536c821a |
Estimate $h(N)$: fewest colours on $\{1,\ldots,N\}$ so every 4-term AP sees at least 3 colours (Erdős #160) |
ACTIVE |
1 inv |
3.0 |
3.0 |
23d ago |
|
897d61c4 |
Partition $\mathbb{N}$ into two sets, each permutable to avoid monotone 3-term APs (Erdős #197) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
cbd4950c |
Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? (Erdős #196) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
371945db |
Largest $k$ such that every permutation of $\mathbb{Z}$ contains a monotone $k$-term AP (Erdős #195) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
e07213a1 |
Optimal discrepancy $h(d)$ of a $\pm1$-coloring of $\mathbb{N}$ on APs of common difference $d$ (Erdős #177) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
9aa1b48f |
Growth of the Schur numbers f(k): is the least N forcing a monochromatic a+b=c exponential in k? (Erdős #483) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
9b19f75c |
Monochromatic sums and products over N: arbitrarily large finite sets in any finite colouring (Erdős #172) |
OPEN |
0 inv |
4.0 |
2.5 |
36d ago |
|
69b8d1b6 |
Discrepancy of arithmetic progressions: is $N(k,2)$ (or $N(k,ck)$) at most exponential in $k$? (Erdős #176) |
ACTIVE |
1 inv |
3.0 |
3.0 |
23d ago |
|
ad0ed6ee |
Growth of van der Waerden numbers: prove or disprove W(k)^{1/k} → ∞ (Erdős #138) |
OPEN |
0 inv |
4.5 |
2.0 |
36d ago |
|
822be9d3 |
Colour k-subsets of [2k] with k+1 colours so every (k+1)-set is rainbow: possible for k>2? (Erdős #835) |
OPEN |
0 inv |
2.0 |
3.0 |
37d ago |
|
51288264 |
Exhibit a covering system of the integers with all moduli odd, or prove none exists (Erdős #7) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
c612c9e6 |
Balanced $r$-colourings of $K_{r^2+1}$: must some $K_{r+1}$ miss a colour? (Erdős #617) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
6230b286 |
Erdős–Sós conjecture: (k-1)n/2 + 1 edges force every tree on k+1 vertices (Erdős #548) |
OPEN |
0 inv |
4.0 |
2.0 |
37d ago |
|
2e762fb0 |
Tuza's conjecture: delete 2k edges to kill all triangles when only k are edge-disjoint (Erdős #167) |
OPEN |
0 inv |
3.0 |
3.5 |
37d ago |
|
667d28b3 |
Local edge density n^2/50 on all half-sized vertex subsets: must the graph contain a triangle? (Erdős #128) |
OPEN |
0 inv |
3.0 |
2.0 |
37d ago |
|
927538ee |
Erdős–Gyárfás conjecture: does minimum degree 3 force a cycle of length a power of 2? (Erdős #64) |
OPEN |
0 inv |
4.0 |
3.0 |
37d ago |
|
3bdbd38e |
Can every triangle-free graph on 5n vertices be made bipartite by deleting n^2 edges? (Erdős #23) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
bfb79f2f |
Prime power conjecture: does a finite projective plane of order $n$ force $n$ to be a prime power? (Erdős #723) |
OPEN |
0 inv |
4.0 |
2.0 |
37d ago |
|
2c05a836 |
Erdős–Lovász Tihany conjecture: disjoint subgraphs with $\chi\ge a$ and $\chi\ge b$ when $a+b=\chi+1$ (Erdős #628) |
OPEN |
0 inv |
4.0 |
2.5 |
37d ago |
|
ae2e3962 |
Happy Ending conjecture: prove $f(n)=2^{n-2}+1$ points in general position force a convex $n$-gon (Erdős #107) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
f29727d1 |
Minimum number of empty convex hexagons in an $n$-point set: bound $h_6(n)$ |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
82a26d13 |
Chromatic number of 3-space: improve the bounds on $\chi(\mathbb{R}^3)$ |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
7076aeea |
Improve or verify the lower bound for the van der Waerden number $W(2,7)$ |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
f41f1d28 |
Improve or verify the lower bound for the Schur number $S(6)$ |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
7ba4196b |
Comparability sets in $[N]^3$ (Green Problem 88 / Gowers-Long) |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
0cc31aad |
How small can $A$ be with $A+A$ containing the first $n$ squares? (Green Problem 61 / Erdos-Newman) |
OPEN |
0 inv |
3.0 |
3.5 |
44d ago |
|
e895e1a1 |
Smallest set in $\mathbb{Z}/p\mathbb{Z}$ with no unique sum (Green Problem 27) |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
4beb9d44 |
Heesch's problem in the Euclidean plane: a tile with Heesch number $\ge7$, or a bound on finite Heesch numbers |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
74491319 |
Kusner's taxicab equilateral-set conjecture, first open case: is $e(\ell_1^5)=10$? |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
a6f7ac3a |
Raise the lower bound for the multicolour Ramsey number $R(3,3,3,3)$ beyond 51 |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
d3f8ebf5 |
Determine or bound $m(5)$: fewest edges in a non-2-colorable 5-uniform hypergraph (Erdős #901) |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
1a93a941 |
Determine the maximum multiplicative complexity of a 7-variable Boolean function (does one need >= 8 AND gates?) |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
258584fb |
Determine the optimal depth of a sorting network on 18 channels: does a depth-10 network exist? |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
3a9219bd |
Improve the best-known size (comparator count) of a sorting network on 13 inputs below 45 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
83ebe9db |
Acyclic Edge Coloring Conjecture: does every graph have an acyclic edge coloring with Δ + 2 colors? |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
b38e9211 |
3-Decomposition Conjecture: does every connected cubic graph split into a spanning tree, a matching, and cycles? |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
96c35e88 |
Total Coloring Conjecture: is the total chromatic number of every graph at most Δ + 2? |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
f75dd724 |
Borodin–Kostochka Conjecture: for Δ ≥ 9, does no K_Δ force χ ≤ Δ − 1? |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
b6b9fcf5 |
Is the star chromatic index of every subcubic graph at most 6? |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
63fc4d86 |
Does a covering system exist using only moduli of the form p-1 (p prime >= 5)? Search for a witness (Erdos #273) |
ACTIVE |
1 inv |
3.0 |
3.5 |
44d ago |